It is often necessary to interface an optical fiber element with an extremely small laser so that light from the laser will enter the element with as little as possible loss due to reflection. For this purpose the optical aperture of the element must be aligned with the laser within .+-.1.mu. normal to the end and 8.mu. axially away from the laser. It has been determined that a tapered end on the element facilitates the coupling efficiency; a preferred taper is effected with a conical tip of 140.degree. to 150.degree., for example. One technique for applying a taper is to heat and draw the tip but this achieves a narrow angle at best, well under 150.degree.. One technique for producing a conical tip with a large enough incident angle is grinding. But an optical fiber element is typically only 0.003 inch in diameter and the light actually only passes through a center section of the element about 8.mu. in diameter. The problem in grinding is that the geometric axis of the conical end must coincide with optical axis of the fiber optic element within about 1.mu.. The optical axis is by definition the center of a light beam exiting the aperture and must be found experimentally. Its location within the 0.003 inch structure cannot be predicted within the required 1.mu. accuracy. Thus the grinding method must include means to determine the location of this axis and then grind the cone onto the fiber centering the cone on this axis.